Data Structures and Algorithms for Engineers
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1 4-63 Data Structures and Algorithms for Engineers David Vernon Carnegie Mellon University Africa wwwvernoneu Data Structures and Algorithms for Engineers 1 Carnegie Mellon University Africa
2 Lecture 18 Graphs Types of graph Adjacency matrix representation Adjacency list representation Breadth-First Search traversal Depth-First Search traversal Topological Sorting Minimum Spanning Tree Prim s Algorithm Kruskal s algorithm Shortest Path Algorithms Dijkstra s algorithm Floyd s algorithm Data Structures and Algorithms for Engineers 2 Carnegie Mellon University Africa
3 Important way of modelling and representing the organization of many systems and problems Road networks Electronic circuits Telecommunication networks Human interaction Social networks Eco-system networks Robot navigation paths Any relationship Data Structures and Algorithms for Engineers 3 Carnegie Mellon University Africa
4 A graph G = (V, E) consists of A set of vertices V A set E of vertex pairs or edges Vertex: node in a graph Edge (arc): a pair of vertices representing a connection between two nodes in a graph Undirected graph: a graph in which the edges have no direction Directed graph (digraph): a graph in which each edge is directed from one vertex to another (or the same) vertex Data Structures and Algorithms for Engineers 4 Carnegie Mellon University Africa
5 The key to solving many algorithmic problems is to think of them in terms of graphs The key to using graphs algorithms effectively in applications is to model your problem correctly to take advantage of existing graph algorithms Data Structures and Algorithms for Engineers 5 Carnegie Mellon University Africa
6 Data Structures and Algorithms for Engineers 6 Carnegie Mellon University Africa
7 Undirected graph G V = {A, B, C, D} E = {(A, B), (A, D), (B, C), (B, D)} Data Structures and Algorithms for Engineers 7 Carnegie Mellon University Africa
8 Directed graph G V = {1, 3, 5, 7, 9, 11} E = {(1, 3), (3, 1), (5, 7), (5, 9), (9, 9), (9, 11), (11, 1)} Data Structures and Algorithms for Engineers 8 Carnegie Mellon University Africa
9 Directed graph G V = {A, B, C, D, E, F, G, H, I, J} E = {(G, D), (G, I), (D, B), (D, F), (I, H), (I, J), (B, A), (B, C), (F, E)} Data Structures and Algorithms for Engineers 9 Carnegie Mellon University Africa
10 Adjacent vertices Two vertices in a graph that are connected by an edge Path A sequence of vertices that connects two nodes in a graph Complete graph A graph in which every vertex is directly connected to every other vertex Weighted graph A graph in which each edge carries a value Data Structures and Algorithms for Engineers 1 Carnegie Mellon University Africa
11 A complete directed graph G Data Structures and Algorithms for Engineers 11 Carnegie Mellon University Africa
12 A complete undirected graph G Data Structures and Algorithms for Engineers 12 Carnegie Mellon University Africa
13 A weighted graph G Data Structures and Algorithms for Engineers 13 Carnegie Mellon University Africa
14 A graph G is undirected if edge (x, y) is an element of E implies (y, x) is an element of E Data Structures and Algorithms for Engineers 14 Carnegie Mellon University Africa
15 For unweighted graphs, the shortest path must have the fewest number of edges and can be found using breadth-first search (see later) Shortest paths in weighted graphs requires more sophisticated algorithms (see later) Data Structures and Algorithms for Engineers 15 Carnegie Mellon University Africa
16 Certain types of edges complicate the task of working with graphs A self-loop is an edge (x, x) involving only one vertex An edge (x, y) is a multiedge if it occurs more than once in the graph Graphs that do not have these types of edges are called simple Data Structures and Algorithms for Engineers 16 Carnegie Mellon University Africa
17 There are n 2 = n! possible vertex pairs in a simple undirected graph with n vertices (n-2)! 2! Graphs where a large fraction of the vertex pairs define edges are called dense Typically dense graphs has a quadratic number of edges, sparse graphs are linear in size Data Structures and Algorithms for Engineers 17 Carnegie Mellon University Africa
18 An acyclic graph does not contain any cycles Trees are connected, acyclic undirected graphs Directed acyclic graphs are called DAGs They arise in scheduling problems where a directed edge (x, y) indicates that activity x must occur before activity y A topological sort orders the vertices of a DAG wrt these precedence constraints Data Structures and Algorithms for Engineers 18 Carnegie Mellon University Africa
19 A graph is embedded if the vertices and edges are assigned geometric positions Data Structures and Algorithms for Engineers 19 Carnegie Mellon University Africa
20 Certain graphs are not explicitly constructed and then traversed, but built as we use them (eg in a backtrack search; see later) Data Structures and Algorithms for Engineers 2 Carnegie Mellon University Africa
21 Each vertex is assigned a unique name in a labelled graph to distinguish it from other vertices In unlabelled graphs, no such distinctions are made Sub-graph isomorphism testing: determine whether the topological structure of two (sub-) graphs are identical if we ignore any labels (typically solved using backtracking, by trying to assign each vertex in each graph a label such that the structures are identical) Data Structures and Algorithms for Engineers 21 Carnegie Mellon University Africa
22 Assuming a graph G = (V, E) with n vertices and m edges, there are two basic choices for data structures Adjacency Matrix: an n n matrix M, where element M[i, j] = 1 if (i, j) is an edge of G, and if it isn t (or, alternatively M[i, j] = w, the weight of the edge) Adjacency List: a linked list that stores the neighbours that are adjacent to each vertex Data Structures and Algorithms for Engineers 22 Carnegie Mellon University Africa
23 Adjacency Matrix for Flight Connections Data Structures and Algorithms for Engineers 23 Carnegie Mellon University Africa
24 Adjacency List for Flight Connections Data Structures and Algorithms for Engineers 24 Carnegie Mellon University Africa
25 While Adjacency Matrices are simpler, Adjacency Lists are the right data structure for most applications of graphs Data Structures and Algorithms for Engineers 25 Carnegie Mellon University Africa
26 /* Adjacency list representation of a graph of degree MAXV */ /* */ /* Directed edge (x, y) is represented by edgenode y in x s */ /* adjacency list Vertices are numbered 1 MAXV */ #define MAXV 1 /* maximum number of vertices */ typedef struct { int y; /* adjacent vertex number */ int weight; /* edge weight, if any */ struct edgenode *next; /* next edge in list */ } edgenode; typedef struct { edgenode *edges[maxv+1]; /* adjacency info: list of edges */ int degree[maxv+1]; /* number of edges for each vertex */ int nvertices; /* number of vertices in graph */ int nedges; /* number of edges in graph */ bool directed; /* is the graph directed? */ } graph; Data Structures and Algorithms for Engineers 26 Carnegie Mellon University Africa
27 degree edges y 1 weight next directed nvertices nedges Data Structures and Algorithms for Engineers 27 Carnegie Mellon University Africa
28 /* Initialize graph from data in a file */ initialize_graph(graph *g, bool directed){ int i; /* counter */ g -> nvertices = ; g -> nedges = ; g -> directed = directed; for (i=1; i<=maxv; i++) g->degree[i] = ; } for (i=1; i<=maxv; i++) g->edges[i] = NULL; Data Structures and Algorithms for Engineers 28 Carnegie Mellon University Africa
29 degree edges false directed nvertices nedges Data Structures and Algorithms for Engineers 29 Carnegie Mellon University Africa
30 /* build graph from data */ read_graph(graph *g, bool directed) { int i; /* counter */ int m; /* number of edges */ int x, y; /* vertices in edge (x,y) */ initialize_graph(g, directed); scanf("%d %d",&(g->nvertices),&m); } for (i=1; i<=m; i++) { scanf("%d %d",&x,&y); insert_edge(g,x,y,directed); } Data Structures and Algorithms for Engineers 3 Carnegie Mellon University Africa
31 /* Initialize graph from data in a file */ insert_edge(graph *g, int x, int y, bool directed) { edgenode *p; /* temporary pointer */ p = malloc(sizeof(edgenode)); /* allocate edgenode storage */ p->weight = ; p->y = y; p->next = g->edges[x]; g->edges[x] = p; /* insert at head of list */ g->degree[x] ++; } if (directed == false) /* NB: if undirected add */ insert_edge(g,y,x,true); /* the reverse edge recursively */ else /* but directed TRUE so we do it */ g->nedges ++; /* only once */ Data Structures and Algorithms for Engineers 31 Carnegie Mellon University Africa
32 insert_edge(g, 1, 2, false) degree edges p false directed 3 nvertices nedges Data Structures and Algorithms for Engineers 32 Carnegie Mellon University Africa
33 /* Initialize graph from data in a file */ insert_edge(graph *g, int x, int y, bool directed) { edgenode *p; /* temporary pointer */ p = malloc(sizeof(edgenode)); /* allocate edgenode storage */ p->weight = ; p->y = y; p->next = g->edges[x]; g->edges[x] = p; /* insert at head of list */ g->degree[x] ++; } if (directed == false) /* NB: if undirected add */ insert_edge(g,y,x,true); /* the reverse edge recursively */ else /* but directed TRUE so we do it */ g->nedges ++; /* only once */ Data Structures and Algorithms for Engineers 33 Carnegie Mellon University Africa
34 insert_edge(g, 1, 2, false) degree edges p false directed 3 nvertices nedges Data Structures and Algorithms for Engineers 34 Carnegie Mellon University Africa
35 /* Initialize graph from data in a file */ insert_edge(graph *g, int x, int y, bool directed) { edgenode *p; /* temporary pointer */ p = malloc(sizeof(edgenode)); /* allocate edgenode storage */ p->weight = ; p->y = y; p->next = g->edges[x]; g->edges[x] = p; /* insert at head of list */ g->degree[x] ++; } if (directed == false) /* NB: if undirected add */ insert_edge(g,y,x,true); /* the reverse edge recursively */ else /* but directed true so we do it */ g->nedges ++; /* only once */ Data Structures and Algorithms for Engineers 35 Carnegie Mellon University Africa
36 insert_edge(g, 2, 1, true) degree edges false directed 3 nvertices nedges Data Structures and Algorithms for Engineers 36 Carnegie Mellon University Africa
37 /* Initialize graph from data in a file */ insert_edge(graph *g, int x, int y, bool directed) { edgenode *p; /* temporary pointer */ p = malloc(sizeof(edgenode)); /* allocate edgenode storage */ p->weight = ; p->y = y; p->next = g->edges[x]; g->edges[x] = p; /* insert at head of list */ g->degree[x] ++; } if (directed == false) /* NB: if undirected add */ insert_edge(g,y,x,true); /* the reverse edge recursively */ else /* but directed true so we do it */ g->nedges ++; /* only once */ Data Structures and Algorithms for Engineers 37 Carnegie Mellon University Africa
38 insert_edge(g, 2, 1, true) degree edges false directed 3 nvertices 1 nedges Data Structures and Algorithms for Engineers 38 Carnegie Mellon University Africa
39 insert_edge(g, 1, 3, false) degree edges false directed 3 nvertices 2 nedges Data Structures and Algorithms for Engineers 39 Carnegie Mellon University Africa
40 /* Print a graph */ print_graph(graph *g) { int i; /* counter */ edgenode *p; /* temporary pointer */ } for (i=1; i<=g->nvertices; i++) { printf("%d: ",i); p = g->edges[i]; while (p!= NULL) { printf(" %d",p->y); p = p->next; } printf("\n"); } Data Structures and Algorithms for Engineers 4 Carnegie Mellon University Africa
41 Consider using a well-established graph library for implementing graph-based applications LEDA Library of Efficient Data types and Algorithms wwwalgorithmic-solutionscom Boost Graph Library wwwboostorg wwwboostorg/libs/graph/doc Data Structures and Algorithms for Engineers 41 Carnegie Mellon University Africa
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